Half-band Filters Research Articles

Algebraic Design of Some Equiripple Linear-Phase Half-Band FIR Filters

Summary A closed form solution for the approximation of a linear-phase FIR (finite impulse response) filter with equiripple magnitude responsein the passband and stopband was not known. In this letter we present a closed form solution of some equiripple linear-phase half-band FIR filter approximation.

Multiplierless interpolator for a delta-sigma digital to analog converter

A simplified algorithm for digital signal interpolation and a novel architecture to implement the algorithm in an integrated circuit ('IC') with significant space constraints are presented. According to the embodiments of the present invention, the interpolator is divided into two parts. The first part of the interpolator increases the sample rate by a factor of two and smoothes the signal using a half-band Infinite Impulse Response ('IIR') filter. The second part of the interpolator increases the sample rate of the signal by a factor of thirty-two using a zero-order-hold ('ZOH') circuit. In one embodiment, the half-band IIR filter is implemented using an all-pass lattice structure to minimize quantization effects. The lattice coefficients are chosen such that the structure can achieve all filter design requirements, yet is capable of being implemented with a small number of shifters and adders, and no multipliers.

Design of low-delay FIR half-band filters with arbitrary flatness and its application to filter banks

Half-band filters are a class of important filters among digital filters, and have been widely used in many applications such as filter banks and wavelets. The conventional methods are mainly concerned with FIR half-band filters with exactly linear phase. However, the exactly linear phase filters have a drawback of large group delay when high-order filters are required. In this paper, the design of FIR half-band filters with a lower group delay is considered. In some applications of filter banks and wavelets, a flat magnitude response is required for the half-band filters. Therefore, a new method for designing low-delay FIR half-band filters with arbitrary flatness is proposed. In the proposed method, while taking the specified flatness condition into account, the design problem is formulated by using the complex Remez exchange algorithm in the stopband. Then, a set of filter coefficients can be easily obtained by solving a simple system of linear equations. The optimal solution with an equiripple response in the stopband is obtained by applying an iteration process. Finally, the proposed method is applied to the design of two-channel perfect reconstruction filter banks with low group delay to demonstrate its effectiveness. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(10): 1–9, 2000

Universal maximally flat lowpass FIR systems

The family of FIR digital filters with maximally flat magnitude and group delay response is considered. The filters were proposed by Baher (1982), who furnished them with an analytic procedure for derivation of their transfer function. The contributions of this paper are the following. A simplified formula is presented for the transfer function of the filters. The equivalence of the novel formula with a formula that is derived from Baher's analytical procedure is proved using a modern method for automatic proof of identities involving binomial coefficients. The universality of Baher's filters is then established by proving that they include linear-phase filters, generalized half-band filters, and fractional delay systems. In this way, several classes of maximally flat filters are unified under a single formula. The generating function of the filters is also derived. This enables us to develop multiplierless cellular array structures for exact realization of a subset of the filters. The subset that enjoys such multiplierless realizations includes linear-phase filters, some nonsymmetric filters, and generalized halfband filters. A procedure for designing the cellular array structures is also presented.

Approximate linear phase multiplierless IIR halfband filter

This paper deals with the real-time multiplierless implementation of halfband IIR filters having approximate linear phase and significantly smaller processing delay than previously reported implementations.

Optical half-band filters

This paper proposes two kinds of novel 2/spl times/2 circuit configuration for finite-impulse response (FIR) half-band filters. These configurations can be transformed into each other by a symmetric transformation and their power transmittance is identical. The configurations have only about half the elements of conventional FIR lattice-form filters. We derive a design algorithm for achieving desired power transmittance spectra. We also describe 2/spl times/2 circuit configurations for infinite-impulse response (IIR) half-band filters. These configurations are designed to realize arbitrary-order IIR half-band filter characteristics by extending the conventional half-band circuit configuration used in millimeter-wave devices. We discuss their filter characteristics and confirm that they have a power half-band property. We demonstrate design examples including FIR maximally flat half-band filters, an FIR Chebyshev half-band filter, and an IIR elliptic half-band filter.

Filter structures composed of all-pass and FIR filters for interpolation and decimation by a factor of two

This paper introduces filter structures for interpolation and decimation by a factor of two. The structures are derived by using the frequency-response masking approach, in which the overall filter makes use of a periodic model filter, its complementary filter, and two masking filters. The periodic model filters are obtained by replacing each delay element in a model filter with M delay elements in cascade. The model filter is a half-band infinite-impulse response (IIR) filter composed of two all-pass filters in parallel, whereas the masking filters are linear-phase finite-impulse response (FIR) filters. In the final interpolator and decimator structures the filtering takes plate at the lowest of the two sampling rates involved. The corresponding overall filter can be designed by separately optimizing a half-band IIR filter and a linear-phase FIR filter. Both nonlinear-phase and approximately linear-phase filters are considered. One advantage of the proposed filter structures over conventional half-band IIR filter structures is that their maximal sample frequency is M times higher, which may be utilized to increase the speed in an implementation and/or to reduce the power consumption via power supply voltage scaling techniques. In the case of approximately linear-phase filters, the computational complexity can be reduced as well. Several design examples are included demonstrating the properties and advantages of the proposed filter structures.

Design and multiplierless realization of maximally flat FIR digital Hilbert transformers

A unified treatment of approximation and realization of type-3 finite impulse response (FIR) linear-phase Hilbert transformers is presented. A simple method based on Bernstein polynomials and half-band filters is proposed to derive the transfer function of the system, and a triangular array realization based on the de Casteljau algorithm is developed from the Bernstein form of the transfer function. It is shown that the array structure, consisting of multiplierless identical modules, can be realized hierarchically using complex and real signal processing techniques.

A method for design of Mth-band filters

The objective of this paper is to present a theory, constraints, and a design method for nonlinear-phase halfband and Mth-band filters. Based on a time-domain property, the constraints and properties in the frequency domain are derived. They are the generalization of the well-known conditions for linear-phase Mth-band filters. Having found all necessary conditions, we present the design method based on an eigenfilter that minimizes the mean-squared errors. The design method is also extended to the design of nonlinear-phase Mth-band filters with properties of R-regularity, or equiripple stopband attenuation, or impulse responses that have complex coefficients. Design examples of various Mth-band filters with different properties are presented, discussed, and compared with the linear-phase case.

On the use of interpolated second-order polynomials for efficient filter design in programmable downconversion

Interpolated second-order polynomials (ISOPs) are proposed to design efficient cascaded integrator-comb (CIC)-based decimation filters for a programmable downconverter. It is shown that some simple ISOPs can effectively reduce the passband droop caused by CIC filtering with little degradation in aliasing attenuation. In addition, ISOPs are shown to be useful for simplifying halfband filters that usually follow CIC filtering. As a result, a modified halfband filter (MHBF) is introduced which is simpler than conventional halfband filters. The proposed decimation filter for programmable downconverter is a cascade of a CIC filter, an ISOP, MHBFs, and a programmable finite impulse response filter. A procedure for designing the decimation filter is developed. In particular, an optimization technique that simultaneously designs the ISOP and programmable FIR filters is presented. Design examples demonstrate that the proposed method leads to more efficient programmable downconverters than existing ones.

High-speed recursive filter structures composed of identical all-pass subfilters for interpolation, decimation, and QMF banks with perfect magnitude reconstruction

High-speed recursive filter structures for interpolation and decimation with factors of two, and quadrature mirror filter (QMF) banks with perfect magnitude reconstruction, are proposed. The structures are composed of identical all-pass subfilters that are interconnected via extra multipliers. For the case of interpolation and decimation filters, the overall transfer function corresponds in the simplest case to several half-band infinite-impulse response (IIR) filters in cascade. To achieve a smaller passband ripple than for a cascade design, a design procedure that has been used earlier for single-rate filters is used. In this approach, the design is split into designs of a prototype finite-impulse response (FIR) filter and a half-band IIR filter. For the case of QMF banks, the design is again separated into designs of a prototype FIR filter and a half-band IIR filter. One major advantage of the proposed filter structures over the corresponding conventional (half-band filter) structures is that the required coefficient word length for the all-pass filters is substantially reduced, implying that the maximal sample frequency can he substantially increased for a given VLSI technology. Further, for interpolation and decimation, the arithmetic complexity may be reduced in comparison with both the conventional structures and straightforward cascade structures. Simple recurrence formulas for computation of the interconnecting multipliers, given the overall transfer function, are derived. Several examples are included which compare the proposed structures with the corresponding conventional and straightforward cascade structures.

Design of biorthogonal FIR linear phase filter banks with structurally perfect reconstruction

In the design of two channel perfect reconstruction filter banks, most of the conventional methods optimize the frequency response of each filter to meet the perfect reconstruction condition. However, quantization of the filter coefficients results in some errors in the frequency response, so it is not guaranteed that the perfect reconstruction condition is still satisfied. In this paper, we present a new method for designing biorthogonal FIR linear phase filter banks with structurally perfect reconstruction. From the perfect reconstruction condition, we first describe a class of structurally perfect reconstruction implementations. Since the proposed filter banks structurally satisfy the perfect reconstruction condition, the design problem becomes the magnitude approximation of the analysis or synthesis filters. Design of these filters can be reduced to the design of half-band filters. We then give a new method to design FIR linear phase half-band filters with arbitrary flatness. Therefore, the proposed filter banks can be designed easily by using the proposed method. Additionally, the magnitude responses of the low- and high-pass filters can be arbitrarily controlled by using two different half-band filters. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(1): 1–8, 1999

Ein analytischer Entwurf von digitalen Halbband-Fittern

A new, simple analytic method for the design of digital half-band filters is derived; for that, we introduce a closed form of the ideal half-band transfer function and apply polynomial approximations to the non-rational part. For the maximally flat half-band filter, we can find a new form especially suited for an implementation without multipliers, often using less than half the number of adders compared with the direct form. Furthermore, half-band filters with nearly perfect Chebyshev approximation can be generated; in general, the ripple exceeds the optimum value by less than 1% and is approximately halved compared to the Kaiser-window method.

Multiplierless and hierarchical structures for maximally flat half-band FIR filters

A simple method to derive a closed-form expression for the transfer function of linear-phase half-band filters with maximally flat amplitude-response characteristics is presented. The method is based on the binomial series. It results in hierarchical and modular structures with low hardware complexity for low-pass and high-pass filters. For a filter of a given order, the structures provide access to ail maximally flat filters of lower orders. Two types of structures are presented. The first enjoys a set of multiplier coefficients with reduced dynamic range, and the second can be realized free of multiplier coefficients in a modular manner. Extension of the order of the filter can be achieved by cascading additional building blocks for the former structure or by adding an extra layer of modules for the latter structure. The proposed structures and formulas are applicable to maximally flat Hilbert transformers as well.

Design of multiplierless elliptic IIR filters with a small quantization error

We present a new technique for the design of multiplierless IIR elliptic filters. The multiplierless filter has all multiplication constants implemented with a small number of shifters and adders. The proposed technique is based on sensitivity analysis. An analytical expression for amplitude response sensitivity is derived for the filter structures consisting of two allpass subfilters in parallel. It is shown that the amplitude response sensitivity to some constant x can be expressed as a product of the filter reflectance function and the phase sensitivity of the allpass section that implements the constant. The closed-form expressions for the phase sensitivities of the first- and second-order allpass sections are also developed. It is shown in the paper that the (n+1)/2 most sensitive constants can be directly controlled by the transfer function parameters if the transfer function is derived by the bilinear transformation from an elliptic minimal Q-factors (EMQF) analog prototype. This way, (n+1)/2 multiplication constants can be implemented without quantization, leaving the filter characteristic strictly elliptic. This is achieved for a class of low-noise allpass sections and for the wave lattice digital filter as well. The quantization of the remaining (n-1)/2 less-sensitive constants is performed using the phase-tolerance scheme and phase-sensitivity functions. The proposed design technique is straight-forward and, consequently, very fast. The application is demonstrated on the examples of narrowband, wideband, and halfband filters.

Adaptive predictor based on maximally flat halfband filter in lifting scheme

For complex short time-varying signals, a high-order predictor does not always yield good performance. For this, we investigate the use of a short-order adaptive predictor. Since the maximally flat filters are the optimal predictors for polynomial signal prediction, the adaptation is based on the combination of a set of maximally flat filters. For compression efficiency, the dynamic ranges of the weighting variables are specially considered. For this, based on the Bernstein filters, another form to represent the weighting variables is used. These two sets of weighting coefficients can be transformed into each other with a simple linear transform. Thus, the adaptation can be made in both the time domain and the frequency domain. For block-based image coding, the least square criterion is used to derive the weighting coefficients. Experimental results show that the adaptive predictor performs better than the S+P transform, the median edge detector (MED), and the gradient adjusted predictor (GAP).

A new approach to synthesize sharp 2D half-band filters

The frequency-response masking method is an efficient technique for the realization of sharp, one-dimensional filters. Recently, this technique has been extended to the synthesis of sharp two-dimensional (2D) filters. While it has been demonstrated that sharp diamond-shaped filters may be realized efficiently, the techniques previously presented cannot be applied to the synthesis of 2D half-band filters. In this brief we propose a modification of the technique for the synthesis of sharp 2D half-band diamond-shaped filters. In addition, by exploiting the special properties of 2D half-band filters, the complexities of the band-edge shaping and masking filters are further reduced. This results in a very efficient implementation of 2D half-band diamond-shaped filters.

Structure and design of two-channel filter banks derived from a triplet of halfband filters

A novel approach for the construction of two-channel one-dimensional (1-D) biorthogonal filter banks is described. This approach is intended to help overcome limitations in an otherwise effective set of recently proposed techniques while retaining the advantages of a simple design and feature-rich structure. The novelty of the method rests on the use of a triplet of halfband fillers, which are combined in a convenient form of the transfer function with adjustable parameters that provide the necessary flexibility in obtaining the desired response. Design procedures are described for the construction of the filter banks by casting the approximation problem in a form that easily lends itself to Remez exchange algorithm. In these design procedures, the analysis and synthesis filters can be either finite-duration impulse response, infinite duration impulse response, or a hybrid. Examples of design of 1-D filter banks and extension of the technique to two-dimensional diamond filter banks are presented. Finally, subband image-coding examples are given to demonstrate the advantage of the proposed filter-bank structure.

Comments on "Subband coding of images using asymmetrical filterbanks

In the above paper, Egger and Li presented a set of two-channel filterbanks, asymmetrical filterbanks (AFB's), for image coding applications. The basic properties of these filters are linear-phase, perfect reconstruction, asymmetric lengths for dual filters, and maximum regularity. In this correspondence, we point out that the proposed AFB's are not new in the sense that the proposed construction is equivalent to the factorization of Lagrange halfband filters, which has been reported by other researchers. In addition, we correct an error in the formulation of constructing AFBs in their paper.

A design method for half-band FIR filters

Present methods for designing half-band filters use iterative approaches such as the well-known Parks-McClellan algorithm or some variant of linear programming. Here we give a direct method using Chebyshev polynomials, which provides a reduction in design time over previous methods by about a factor of ten. For all practical purposes the response of the resulting filter is equal-ripple in both passband and stopband. A listing of a Fortran program for carrying out the design is included.